I know the traditional explanation, which says that ice has large spaces between $\ce{H2O}$ molecules because hydrogen-bonding gives it an open structure. But what does the open structure have to do with hydrogen-bonding? Why isn't a similar phenomenon observed in other species which exhibit hydrogen-bonding, like $\ce{HF}$ or $\ce{NH3}$?

$\begingroup$This certainly doesn't answer your question, but FYI it's a myth that expansion upon freezing is unique to water. There are any number of organic compounds for which this is the case, also some pure elements which clearly to not hydrogen bond like gallium, antimony, germanium, silicon and more.$\endgroup$
– airhuffApr 30 '17 at 17:36

$\begingroup$What I don't get is why the hydrogen bonding somehow causes the ice to have an open structure.$\endgroup$
– Saad Apr 30 '17 at 20:00

Note that the dashed lines represent hydrogen bonds. Liquid water actually has a similar "open" structure also due to hydrogen bonding. But in the case of liquid water, the hydrogen bonds are not rigid and semi-permanent as in ice. So imagine that in the image above, the hydrogen bonding network collapses. This is what happens when enough thermal energy is present to break the rigid hydrogen bonds resulting in melting. Clearly, once this crystaline structure is no longer forced into place by the rigid hydrogen bonding in ice, it can collapse into itself, resulting a greater density of water molecules.

Thus the liquid form of water, although engaged in transient hydrogen bonding, is not as open and expanded as when held into it's solid form by the rigid, semi-permanent hydrogen bonding.

As an addition to the other answers, note that the "honeycomb" structure that is responsible for the lowered density upon freezing is not sacrosanct. It can be collapsed, without melting, by high pressure starting at about 200 MPa. These are the high pressure ice phases, about a dozen of which are known. All those in equilibrium with the liquid, apart from the low pressure Ice $I_h$ phase, are more dense than the liquid with which they are in equibrium; so the melting point of water starts rising once we get to the threshold of Ice $III$ at about -22°C and 210 MPa.

We can also go the other way, generating ice structures even more open and lower in density than Ice $I_h$. Such phases are not realized in pure water but appear in clathrates such as the well known methane clathrate.